# Are Continuum Predictions of Clustering Chaotic?

## TL;DR

This paper demonstrates that continuum models of gas-solid clustering in multiphase flows exhibit chaotic behavior, characterized by sensitive dependence on initial conditions and positive Lyapunov exponents, with chaos being dimensionally dependent.

## Contribution

It provides evidence that continuum predictions of clustering are chaotic and links this chaos to secondary multidimensional instabilities rather than fundamental kinematic instability.

## Key findings

- System exhibits sensitive dependence on initial conditions.
- Largest Lyapunov exponent is positive, indicating chaos.
- Chaotic behavior depends on the system's dimensionality.

## Abstract

Gas-solid multiphase flows are prone to develop an instability known as clustering. Two-fluid models, which treat the particulate phase as a continuum, are known to reproduce the qualitative features of this instability, producing highly-dynamic, spatiotemporal patterns. However, it is unknown whether such simulations are truly aperiodic or a type of complex periodic behavior. By showing that the system possesses a sensitive dependence on initial conditions and a positive largest Lyapunov exponent, $\lambda_1 \approx 1/\tau$, we provide a tentative answer: continuum predictions of clustering are chaotic. We further demonstrate that the chaotic behavior is dimensionally dependent, a conclusion which unifies previous results and strongly suggests that the chaotic behavior is not a result of the fundamental kinematic instability, but of the secondary (inherently multidimensional) instability.